It is instructive to study the modulation of one sinusoid by
another. In this section, we will look at sinusoidal Amplitude
Modulation (AM). The general AM formula is given by

where
are parameters of the sinusoidal
carrier wave,
is called the
modulation index (or AM index),
and
is the amplitude modulation signal. In
AM radio broadcasts, is the audio signal being transmitted
(usually bandlimited to less than 10 kHz), and is the channel
center frequency that one dials up on a radio receiver.
The modulated signal
can be written as the sum of the
unmodulated carrier wave plus the product of the carrier wave and the
modulating wave:

(4.1)

In the case of sinusoidal AM, we have

(4.2)

Periodic amplitude modulation of this nature is often called the
tremolo effect when
or so ( Hz).

Let's analyze the second term of Eq.(4.1) for the case of sinusoidal
AM with and
:

(4.3)

An example waveform is shown in Fig.4.11 for Hz and
Hz. Such a signal may be produced on an analog synthesizer
by feeding two differently tuned sinusoids to a ring modulator,
which is simply a ``four-quadrant multiplier'' for analog signals.

When is small (say less than radians per second, or
10 Hz), the signal is heard as a ``beating sine wave'' with
beats per second. The beat rate is
twice the modulation frequency because both the positive and negative
peaks of the modulating sinusoid cause an ``amplitude swell'' in
. (One period of modulation-- seconds--is shown in
Fig.4.11.) The sign inversion during the negative peaks is not
normally audible.

Recall the trigonometric identity for a sum of angles:

Subtracting this from
leads to the identity

Setting
and
gives us an alternate form
for our ``ring-modulator output signal'':

(4.4)

These two sinusoidal components at the sum and difference
frequencies of the modulator and carrier are called
side bands
of the carrier wave at frequency (since typically
).

Equation (4.3) expresses as a ``beating sinusoid'', while
Eq.(4.4) expresses as it two unmodulated sinusoids at
frequencies
. Which case do we hear?

It turns out we hear as two separate tones (Eq.(4.4))
whenever the side bands are resolved by the ear. As
mentioned in §4.1.2,
the ear performs a ``short time Fourier analysis'' of incoming sound
(the basilar membrane in the cochlea acts as a mechanical
filter bank). The
resolution of this filterbank--its ability to discern two
separate spectral peaks for two sinusoids closely spaced in
frequency--is determined by the
critical bandwidth of hearing
[45,76,87]. A critical
bandwidth is roughly 15-20% of the band's center-frequency, over most
of the audio range [71]. Thus, the side bands in
sinusoidal AM are heard as separate tones when they are both in the
audio range and separated by at least one critical bandwidth. When
they are well inside the same critical band, ``beating'' is heard. In
between these extremes, near separation by a critical-band, the
sensation is often described as ``roughness'' [29].

Equation (4.4) can be used to write down the spectral representation of
by inspection, as shown in Fig.4.12. In the example
of Fig.4.12, we have Hz and Hz,
where, as always,
. For comparison, the spectral
magnitude of an unmodulated Hz sinusoid is shown in
Fig.4.6. Note in Fig.4.12 how each of the two
sinusoidal components at Hz have been ``split'' into two
``side bands'', one Hz higher and the other Hz lower, that
is,
. Note also how the
amplitude of the split component is divided equally among its
two side bands.

figure[htbp]

Recall that was defined as the second term of
Eq.(4.1). The first term is simply the original unmodulated
signal. Therefore, we have effectively been considering AM with a
``very large'' modulation index. In the more general case of
Eq.(4.1) with given by Eq.(4.2), the magnitude of
the spectral representation appears as shown in Fig.4.13.